Canonical Lie-transform method in Hamiltonian gyrokinetics: a new approach

The well-known gyrokinetic problem regards the perturbative expansion related to the dynamics of a charged particle subject to fast gyration motion due to the presence of a strong magnetic field. Although a variety of approaches have been formulated in the past to this well known problem, surprisingly a purely canonical approach based on Lie transform methods is still missing. This paper aims to fill in this gap and provide at the same time new insight in Lie-transform approaches. INTRODUCTION: TRANSFORMATION APPROACH TO GYROKINETIC THEORY A great interest for the description of plasmas is still vivid in the scientific community. Plasmas enter problems related to several fields from astrophysics to fusion theory. A crucial and for some aspects still open theoretical problem is the gyrokinetic theory, which concerns the description of the dynamics for a charged point particle immersed in a suitably intense magnetic field. In particular, the “gyrokinetic problem” deals with the construction of appropriate perturbation theories for the particle equations of motion, subject to a variety of possible physical conditions. Historically, after initial pioneering work [1, 2, 3], and a variety of different perturbative schemes, a general formulation of gyrokinetic theory valid from a modern perspective is probably due to Littlejohn [4], based on Lie transform perturbation methods [5, 6, 7, 8]. For the sake of clarity these gyrokinetic approaches can be conveniently classified as follows (see also Fig.1): A) direct non-canonical transformation methods: in which non-canonical gyrokinetic variables are constructed by means of suitable one-step [1], or iterative, transformation schemes, such as a suitable averaging technique [9], a one-step gyrokinetic transformation [10], a non-canonical iterative scheme [11]. These methods are typically difficult (or even impossible) to be implemented at higher orders; B) canonical transformation method based on mixed-variable generating functions: this method, based on canonical perturbation theory, was first introduced by Gardner [2, 12] and later used by other authors [13]). This method requires, preliminarily, to represent the Hamiltonian in terms of suitable field-related canonical coordinates, i.e., coordinates depending on the the topology of the magnetic flux lines. This feature, added to the unsystematic character of canonical perturbation theory, makes its application to gyrokinetic theory difficult, a feature that becomes even more critical for higher-order perturbative calculations; C) non-canonical Lie-transform methods: these are based on the adoption of the non-canonical Lie-transform perturbative approach developed by Littlejohn [4]. The method is based on the use arbitrary non-canonical variables, which can be field-independent. This feature makes the application of the method very efficient and, due to the peculiar features for the perturbative scheme, it permits the systematic evaluation of higher-order perturbative terms. The method has been applied since to gyrokinetic theory by several authors [14, 15, 16, 17]; D) canonical Lie-transform methods applied to non-canonical variables: see for example [18]. Up to now this 1 [email protected] 2 [email protected] 3 Web site: http://cmfd.univ.trieste.it FIGURE 1. The transformation approach to gyrokinetic theory: 2 : see Gardner[2],[12]; 3 : see Hahm et al. [18] and present theory; 4 : first obtained by Alfven [1]; 5 : see Littlejohn [4]. method has been adopted in gyrokinetic theory only using preliminar non-canonical variables, i.e., representing the Hamiltonian function in terms of suitable, non-canonical variables (similar to those adopted by Littlejohn). This method, although conceptually similar to the developed by Littlejohn, is more difficult to implement. All of these methods share some common features,in particular: they may require the application of multiple transformations, in order to construct the gyrokinetic variables; the application of perturbation methods requires typically the representation of the particle state in terms of suitable, generally non-canonical, state variables. This task may be, by itself, difficult since it may require the adoption of a preliminary perturbative expansion. An additional important issue is the construction of gyrokinetic canonical variables. The possibility of constructing canonical gyrokinetic variables has relied, up to now, on essentially two methods, i.e., either by adopting a purely canonical approach, like the one developed by Gardner [2, 12], or using the so-called “Darboux reduction algorithm”, based on Darboux theorem [4]. The latter is obtained by a suitable combination of dynamical gauge and coordinate transformations, permitting the representation of the fundamental gyrokinetic canonical 1-form in terms of the canonical variables. The application of both methods is nontrivial, especially for higher order pertubative calculations. The second method, in particular, results inconvenient since it may require an additional perturbative sub-expansion for the explicit evaluation of gyrokinetic canonical variables. For these reasons a direct approach to gyrokinetic theory, based on the use of purely canonical variables and transformations may result a viable alternative. Purpose of this work is to formulate a “purely” canonical Lie-transform theory and to explicitly evaluate the canonical Lie-generating function providing the canonical gyrokinetic transformation. LIE-TRASFORM PERTURBATION THEORY We review some basic aspects of perturbation theory for classical dynamical systems. Let us consider the state x of a dynamical system and its d-dimensional phase-space M endowed with a vector field X . With respect to some variables x = { xi } we assume that X has representation [6] dxi dt = X i (1) where ε is an ordering parameter. We treat all power series formally; convergence is of secondary concern to us. By hypothesis, the leading term X0 of (1) represents a solvable system, so that the integral curves of X are approximated by the known integral curves of X0. The strategy of perturbation theory is to seek a coordinate transformation to a new set of variables { x̄i } , such that with respect to them the new equations of motion are simplified. Since (1) is solvable at the lowest order, the coordinate transformation is the identity at lowest order, namely x̄ = x + O(ε) (2) The transformation is canonical if it preserves the fundamental Poisson brackets. It can be determined by means of generating functions, Lie generating function or mixed-variables generating functions, depending on the case. In the Lie transform method, one uses transformations T which are represented as exponentials of some vector field, or rather compositions of such transformations. To begin, let us consider a vector field G, which is associated with the system of ordinary differential equations dxi dε = G(x), (3) so that if x and x̄ are initial and final points along an integral curve (3), separated by an elapsed parameter ε , then x̄ = Tx. In the usual exponential representation for advance maps, we have T = exp(εG). (4) We will call G the generator of the transformation T . In Hamiltonian perturbation theory the transformation T is usually required to be a canonical transformation. Canonical transformations have the virtue that they preserve the form of Hamilton’s equations of motion. Canonical transformation can be represented by mixed-variable generating function, as in the Poincare-Von Zeipel method or by means of Lie transform. In the latter method vector fields G are specified through the Hamilton’s equations. Following a more conventional approach, we can write the (3) in terms of the transformed point dx̄ dε = [x̄,ω ] (5) The components of the above relation are just Hamilton’s equations in Poisson bracket notation applied to the “Hamiltonian” (Lie generating function) ω , with the parameter ε the “time.” Equation (5) therefore generates a canonical transformation for any ε to a final state x̄ whose components satisfy the Poisson bracket condition [q̄i, q̄ j] = [p̄i, p̄ j] = 0 (6) [q̄i, p̄ j] = δi j. (7) To find the transformation T explicitly, we introduce the Lie operator L = [ω , . . . ]. Recalling that coordinate components of vector are subject to pull back transformation law, then one gets dT dε = −TL (8) with the formal solution T = exp [